ilifornia 

;ional 

ility 


UNIVERSITY  OF  CALIFORNIA 


I 


CLASS-ROOM  NOTES  ON  UNIPLANAR  KINEMATICS 


•      4       e 


r* 


mitiMl 

AA      Sciences 
^  ^       Library 

UNIVERSITY  OF  CALIFORNIA. 


CLASS-ROOM  NOTES  ON  UNIPLANAR  KINEMATICS. 

Velocity  in  a  Plane.  A  point  moving  in  a  plane  is  at  any 
instant  determined  in  position  by  a  pair  of  co-ordinates  x,  y  or 
r,  rV,  and  its  path  may  usually  be  denned  by  an  equation, 
y  =  /(x),  or  r  =  <p  (rV).  If  p  be  the  vector  (directed  magnitude) 
from  the  origin  to  the  point  x,  y  or  r,  ft,  this  path  is  more 
completely  defined  by  the  equations 

P  =  x  +  iy,    y  ==/(#), 

involving  a  complex  function  of  the  real  variable  x,  or  also,  in 
a        terms  of  tensor  and"  amplitude,  by  the  equations 

^V  p  =  r  cis  rV,     r  =  <p  (0). 

""^i  The  derivative  of  this  vector  p  with  respect  to  an  equicres- 

*  cent  variable  t,  which  may  be  taken  to  represent  time,  is  its  rate, 

^  and  is  defined  as  the  velocity  of  the  moving  point  whose  position 

4  at  any  instant  it  determines.     In  terms  of  x  and  y  this  velocity  is 

**^  dp       dx         .  dy 

di  " dt   +  l   di 

dx 
and   has  a  component     ,-■    in    the    direction    of   the    ^-axis,    a 


V 


v3^  component    -j   in  the  direction  of  the  jy-axis,    and  the  speed  of 
the  point  in  its  path  is  the  tensor  of  this  velocity,  or 


dp  |  / 1  dx  \ 

dt 


462620        Mi 


UNIPLANAR    KINEMATICS. 

On  the  other  hand,  in  terras  of  r  and  #,  the  velocity  is 


dp 

It  ~~ 

and  its  components  are 
dr 


dr  .    adH 


dt 


cis  0  ==  velocity  in  the  direction  of  the  vector  p 
dH 


ir  cis  B  —  =  velocity  in  a  direction  perpendicular  to  p, 
and  the  speed  of  the  point  in  its  path  is 


dp 
dt 


v«r+-{ 


rdB\ 

dt  S 


The  components  of  velocity  in  two  directions  perpendicular 
to  each  other  as  here  determined  are  independent  of  each  other 
in  the  sense  that  neither  involves  any  motion  whatever  in  the 
direction  of  the  other.  Obviously  this  is  the  case  whatever  be 
the  mode  of  resolution  into  two  rectangular  directions. 

Velocity  is  compounded  in  still  another  important  way  by 
introducing  a  second  variable  s,  assumed  to  represent  the  length 
of  the  path  of  the  moving  point  measured  from  a  fixed  point. 
In  this  form  it  is 

dp ds  dp 

dt         dt  ds 
=  v  p', 


where  v 


ds 
dt 


and  p' 


dp 
ds 


,  and  since  ds 2  =  dx1  -f-  dy1, 


>/{£}+ If} 


That    I  p1 


i  appears  at  once  from  the  fact  that 


and 


dp  dx         .dy 

ds         ds  ds 


dp     2  =      f  dx  | ' 
ds  ids  i     + 


{*}'-'■ 


UNIPLANAR    KINEMATICS. 


Hence  v  is  the  speed  of  the  point  along  its  path,  as  was  asserted 
in  the  equation  giving  -=-  above,  and  p'  is  the  velocity- 
direction,  as  is  otherwise  evident  from  the  fact  that    .  |   /[ 

As=o  (  As  ) 

has  the  direction  of  a  tangent  to  the  path  of  the  point  at  the 
instant  considered. 

Acceleration.      The   rate    of   change   of  velocity   is   called 
acceleratio7i.     Expressed  as  a  derivative  it  is 


d2p d     ids  dp 

W "'"  It  \dids 


I 


d2s  dp        d2p  (ds 
=  df%ds  +  dF\Jt 

■  r  ds  dp  d2p 

orif  as  — T^*  sf  -  "- 

d'fi        dv    ,         ,    „ 

,tls  dF=  <&?  +  "»■ 

and,  like  the  analogous  expressions  for  velocity,  has  two  independ- 

dv 
ent  components   ,-  p'  and  i?p" ,  the  former  in  the  direction  of  the 

tangent  and  in  magnitude  equal  to  the  rate  of  change  of  speed, 
the  latter,  as  will  be  shown  in  the  next  paragraph,  in  a  direction 
perpendicular  to  the  tangent  and  equal  in  magnitude  to  the 
square  of  the  speed  multiplied  by  the  curvature  of  the  path  at 
the  point  considered.  The  first  is  called  tangenital,  the  second 
normal  acceleration. 

Radius  of  Curvature.     L,et  the  equations  of  a  plane  curve  be 
P  =  x  +  iy,    y  =/(#), 

and  let  differentiation  with  respect  to  the  arc  s,  estimated  from  a 

dp  d2x 

fixed  point,   be  denoted   by  accents,     ,     =  p ,    tt  =  -* ',    etc. 

as  dsr 

Then 


462620 


4  UNIP^ANAR   KINEMATICS. 

p'    =  X?    +   *'/ 

p"  =  x"  —  iy'\ 
and  since      \  p\  =■  \ 

.-.   x?2  +  y2  =  i 

and  #V+yy=o, 

nC  p"  x"'2  +  r"2 

*  (^y  — jtry ) 

*"2  +  y2  » 

which  shows  that  p"  is  perpendicular  to  p',   for  i  used    as   a 
multiplier  turns  any  line  in  the  plane  through  a  right  angle. 
And  since    |  p'  |  =  i  and    |  p"  |  "==  .r"2  +  y 2, 
...    |p"|  =  |*"y  — *y»  |  . 

But  if  cp  be  the  angle  between  the  jr-axis  and  the  tangent  at  the 

extremity  of  p 

limit  Ax         ,  .  limit  Ay 

.         —r-  =  x  =  cos  q>,    and    .  -p-  =  y  —  sin  <z>, 

or  these  expressions  for  sin  <p  and  cos  qj  may  be  deduced  from 
the  equation  (dx/ds)2-\-  (dy/ds)2—  i  by  assuming  dx/ds  =  cos  cp, 
inferring  therefrom  i  —  (dx/ds)2  =  (dy/ds)2  =  sin2  (p  and  identi- 
fying cp  as  the  angle  named.    Then,  by  a  second  differentiation, 

x '—  —  sin  q>  — ,  y  =  cos  q>  ~- , 
ds  ds 

and  thence,  by  cross  multiplication  with  the  previous  equations, 

x'y" —  x"y=  (cos2  q>  4-  sin2  cp)   -~  =  —-. 

ds       ds 

d<p 

is  called  the  curvature  of  the  curve  at  the  point  whose  vector 

is  p.  Thus  the  assertion  concerning  the  magnitude  of  the  second, 
or  normal  component  of  acceleration,  at  the  close  of  the  last 
article,  is  verified. 

The  radius  of  curvature  is  the  reciprocal  of  the  curvature,  or 

R  =  %.  =  ±   -'»  ■ 

dcp  fj" 


UNIPLANAR    KINEMATICS.  5 

Problems.  Verify  the  following  formulae  for  the  deter- 
mination of  velocity  and  acceleration. 

(i).   If  a  point  move  with  speed  v  in  the  curve  y  =  fix), 
represented  in  Cartesian  co-ordinates,  prove  that  its  velocity  is 
dp  _  v[i  +  if  {x)\ 
dt~     i     x  +  [/'<*)]' 
(2).  If  a  point  move  with  velocity  v  in  the  curve  r  =f  (6), 
represented  in  polar  co-ordinates,  prove  that  its  velocity  is 

dp  =  vif'jff)  +rf/(fl]pl.fl 

(3).  If  v  =  speed,  and  R  =  radius  of  curvature,  prove  that 
the  tensor  of  acceleration  is 

d*fi     _     \\dvV1      v* 
dt%  \\dt\  ^  Rl- 

In  the  following  curves  let  5  =  length  of  curve,  #  =  vectorial 
angle,  x  =  abscissa  in  a  Cartesian  system.  If  in  each  a  point 
move  with  speed  v,  determine  the  expressions  for  velocity, 
acceleration,   and  radius  of  curvature. 

(4.)  p  =  -|/tf2  -\-  sl  +  ia  sinh  -. 

dp  _     v  (s  -f-  /a)       rf'2p  _      z>2  (a2 —  /as) 
dt  ~       y  J+IT  '    #*"  =       "C^T^OF  ' 

angular  spiral. 

afp         v  (1  -(-  in) 


(5)  p  =  ae        cis  0,     the  equiangular  spiral 


CIS 


0. 


dt  ~ '  V  1  +  »* 
(6).  p«#  cis  8,    the  spiral  of  Archimedes. 

(7)-  Pit  Cls  &t    the  reciprocal  spiral. 

(8).  p  =  x  +  ?'«^v",     the  logarithmic  curve. 

x 
(9).  p  =  x  -j-  zV  cosh  — ,     the  catenary. 

Irving  Stringham. 

/j/7*  February,  1893. 


SOUTHERN  REGIONAL  LIBRARY  FACILITY 

405  Hllgard  Avenue,  Los  Angeles,  CA  90024-1388 

Return  this  material  to  tbe  library 

from  which  it  was  borrowed. 


QL    APR 


19199J 


f .7  R  C  C  1993 
ART  LIBRARY 


J 


lm 


m  '■ 


University 

Southei 

Librai 


